[tr352]

Fuzzy logic deals with this in the exact same way as classical logic. Note that in classical logic, the truth value of "A and B" and "A or B" is also a function of the truth values of A and of B. When using fuzzy logic you have to make a choice as to which function you use. Typically these functions generalise the classical logic ones in the sense that they behave like the classical ones when using 0 and 1. These functions are defined by a so called T-norm.

[lonelappde]

That's great in toy theory, like the formal real number system.

How useful is it for modeling real world problem?

[jacques_chester]

Very useful. Many problems are more easily represented as fuzzy sets or fuzzy relations than in other terms. It's particularly good at encoding linguistic variables, such as "very fast", "too cold", "accelerate hard" and so on in a way that it can smoothly overlap them.

It's also useful for encoding uncertainties that are not yet mutually exclusive. There are other logics too (eg Dempster-Shafer evidence theory), often grouped together as "monotone measures".